docs: def orifice_expansibility_1989(D, Do, P1, P2, k):
      r'''Calculates the expansibility factor for orifice plate calculations
      based on the geometry of the plate, measured pressures of the orifice, and
      the isentropic exponent of the fluid.
      
      .. math::
          \epsilon = 1- (0.41 + 0.35\beta^4)\Delta P/\kappa/P_1
          
      Parameters
      ----------
      D : float
          Upstream internal pipe diameter, [m]
      Do : float
          Diameter of orifice at flow conditions, [m]
      P1 : float
          Static pressure of fluid upstream of orifice at the cross-section of
          the pressure tap, [Pa]
      P2 : float
          Static pressure of fluid downstream of orifice at the cross-section of
          the pressure tap, [Pa]
      k : float
          Isentropic exponent of fluid, [-]
  
      Returns
      -------
      expansibility : float
          Expansibility factor (1 for incompressible fluids, less than 1 for
          real fluids), [-]
  
      Notes
      -----
      This formula was determined for the range of P2/P1 >= 0.75, and for fluids
      of air, steam, and natural gas. However, there is no objection to using
      it for other fluids.
      
      This is an older formula used to calculate expansibility factors for 
      orifice plates.
      
      In this standard, an expansibility factor formula transformation in terms  
      of the pressure after the orifice is presented as well. This is the more
      standard formulation in terms of the upstream conditions. The other formula
      is below for reference only:
      
      .. math::
          \epsilon_2 = \sqrt{1 + \frac{\Delta P}{P_2}} -  (0.41 + 0.35\beta^4)
          \frac{\Delta P}{\kappa P_2 \sqrt{1 + \frac{\Delta P}{P_2}}}
          
      [2]_ recommends this formulation for wedge meters as well.
  
      Examples
      --------
      >>> orifice_expansibility_1989(D=0.0739, Do=0.0222, P1=1E5, P2=9.9E4, k=1.4)
      0.9970510687411718
  
      References
      ----------
      .. [1] American Society of Mechanical Engineers. MFC-3M-1989 Measurement 
         Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2005.
      .. [2] Miller, Richard W. Flow Measurement Engineering Handbook. 3rd
         edition. New York: McGraw-Hill Education, 1996.
      '''
      return 1.0 - (0.41 + 0.35*(Do/D)**4)*(P1 - P2)/(k*P1)